We compute the integral Picard group of the stack K2lβ of
quasi-polarized K3 surfaces of degree 2l=4,6,8. We show that in this range
the integral Picard group is torsion-free and that a basis is given by certain
elliptic Noether-Lefschetz divisors together with the Hodge line bundle. To
achieve this result, we investigate certain stacks of complete intersections
and their Picard groups by means of equivariant geometry. In the end we compute
an expression of the class of some Noether-Lefschetz divisors, restricted to an
open substack of K2lβ, in terms of the basis mentioned above.Comment: 24 pages, comments are welcome